Handbook of Solvents - George Wypych - ChemTech - Ventech!

480 Jacopo Tomasi, Benedetta Mennucci, Chiara Cappelli
Trying to give a more theoretically-stated definition of continuum models, but still
keeping the same conciseness of the original one reported above, we have to introduce some
basic notions of classical electrostatics.
The electrostatic problem of a charge distribution ρ M (representing the molecular solute)
contained in a finite volume (the molecular cavity) within a continuum dielectric can be
expressed in terms of the Poisson equation:
→�
� � � �
∇ ∇ =−4
[8.115]
[ ε() r φ() r ] πρM()
r
where φis the total electrostatic potential.
Eq. [8.115] is subject to the constraint that the dielectric constant inside the cavity is
εin=1, while at the asymptotic boundary we have:
�
2 �
limrφ
r = α limr
φ r = β
[8.116]
r→∞
()
r→∞
()
with α and β finite quantities.
At the cavity boundary the following conditions hold:
� � ∂φ �
in ∂φ
φin() r = φout()
r = ε out()
r
∂n�
∂n�
out
[8.117]
where subscripts in and out indicate functions defined inside and outside the cavity, and �n is
the unit vector perpendicular to the cavity surface and pointing outwards.
Further conditions can be added to the problem, and/or those shown above (e.g., the
last one) can be modified; we shall consider some of these special cases later.
As said above, in several models an important simplification is usually introduced,
i.e., the function ε out (r)
� is replaced by a constant ε (from now on we skip the redundant subscript
out). With this simplification we may rewrite the electrostatic problem by the following
equations:
2 � �
2
inside the cavity: ∇ φ() r =−4πρM
() r , outside the cavity: ∇ φ() = 0
� r [8.118]
Many alternative approaches have been formulated to solve this problem.
In biochemistry, methods that discretize the Poisson differential operator over finite
elements (FE) have long been in use. In general, FEM approaches do not directly use the
molecular cavity surface. Nevertheless, as the whole space filled by the continuous medium
is partitioned into locally homogeneous regions, a careful consideration of the portion of
space occupied by the molecular solute has still to be performed.
There is another family of methods, known as FD (finite difference) methods, 85 which
exploit point grids covering the whole space, but conversely to the FEM point, in FD methods
the points are used to replace differential equations by algebraic ones.
These methods can be applied both to the “simplified” model with constant ε, and to
�
models with space dependent ε( r) or real charges dispersed in the whole dielectric medium.
They aim at solving the Poisson equation (1) expressed as a set of finite difference equations
for each point of the grid. The linear system to be solved has elements depending both on �